3.6.99 \(\int (c x)^{3/2} (a+b x^2)^{3/2} \, dx\) [599]

Optimal. Leaf size=181 \[ \frac {8 a^2 c \sqrt {c x} \sqrt {a+b x^2}}{77 b}+\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}-\frac {4 a^{11/4} c^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a+b x^2}} \]

[Out]

2/11*(c*x)^(5/2)*(b*x^2+a)^(3/2)/c+12/77*a*(c*x)^(5/2)*(b*x^2+a)^(1/2)/c+8/77*a^2*c*(c*x)^(1/2)*(b*x^2+a)^(1/2
)/b-4/77*a^(11/4)*c^(3/2)*(cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c
*x)^(1/2)/a^(1/4)/c^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))*(a^(1/2)
+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(5/4)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {285, 327, 335, 226} \begin {gather*} -\frac {4 a^{11/4} c^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a+b x^2}}+\frac {8 a^2 c \sqrt {c x} \sqrt {a+b x^2}}{77 b}+\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^(3/2)*(a + b*x^2)^(3/2),x]

[Out]

(8*a^2*c*Sqrt[c*x]*Sqrt[a + b*x^2])/(77*b) + (12*a*(c*x)^(5/2)*Sqrt[a + b*x^2])/(77*c) + (2*(c*x)^(5/2)*(a + b
*x^2)^(3/2))/(11*c) - (4*a^(11/4)*c^(3/2)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*Elli
pticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(77*b^(5/4)*Sqrt[a + b*x^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int (c x)^{3/2} \left (a+b x^2\right )^{3/2} \, dx &=\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}+\frac {1}{11} (6 a) \int (c x)^{3/2} \sqrt {a+b x^2} \, dx\\ &=\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}+\frac {1}{77} \left (12 a^2\right ) \int \frac {(c x)^{3/2}}{\sqrt {a+b x^2}} \, dx\\ &=\frac {8 a^2 c \sqrt {c x} \sqrt {a+b x^2}}{77 b}+\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}-\frac {\left (4 a^3 c^2\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{77 b}\\ &=\frac {8 a^2 c \sqrt {c x} \sqrt {a+b x^2}}{77 b}+\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}-\frac {\left (8 a^3 c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{77 b}\\ &=\frac {8 a^2 c \sqrt {c x} \sqrt {a+b x^2}}{77 b}+\frac {12 a (c x)^{5/2} \sqrt {a+b x^2}}{77 c}+\frac {2 (c x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 c}-\frac {4 a^{11/4} c^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{77 b^{5/4} \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.05, size = 89, normalized size = 0.49 \begin {gather*} \frac {2 c \sqrt {c x} \sqrt {a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}-a^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{11 b \sqrt {1+\frac {b x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(3/2)*(a + b*x^2)^(3/2),x]

[Out]

(2*c*Sqrt[c*x]*Sqrt[a + b*x^2]*((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] - a^2*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b
*x^2)/a)]))/(11*b*Sqrt[1 + (b*x^2)/a])

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Maple [A]
time = 0.04, size = 150, normalized size = 0.83

method	result	size
default	\(-\frac {2 c \sqrt {c x}\, \left (-7 b^{4} x^{7}+2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a^{3}-20 a \,b^{3} x^{5}-17 a^{2} b^{2} x^{3}-4 a^{3} b x \right )}{77 x \sqrt {b \,x^{2}+a}\, b^{2}}\)	\(150\)
risch	\(\frac {2 \left (7 b^{2} x^{4}+13 a b \,x^{2}+4 a^{2}\right ) x \sqrt {b \,x^{2}+a}\, c^{2}}{77 b \sqrt {c x}}-\frac {4 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) c^{2} \sqrt {c x \left (b \,x^{2}+a \right )}}{77 b^{2} \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(188\)
elliptic	\(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 b c \,x^{4} \sqrt {b c \,x^{3}+a c x}}{11}+\frac {26 a c \,x^{2} \sqrt {b c \,x^{3}+a c x}}{77}+\frac {8 a^{2} c \sqrt {b c \,x^{3}+a c x}}{77 b}-\frac {4 a^{3} c^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 b^{2} \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\)	\(213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(3/2)*(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/77*c/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)*(-7*b^4*x^7+2*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2
))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)
)*2^(1/2)*(-a*b)^(1/2)*a^3-20*a*b^3*x^5-17*a^2*b^2*x^3-4*a^3*b*x)/b^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(3/2)*(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)*(c*x)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.32, size = 69, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\left (4 \, \sqrt {b c} a^{3} c {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (7 \, b^{3} c x^{4} + 13 \, a b^{2} c x^{2} + 4 \, a^{2} b c\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{77 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(3/2)*(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-2/77*(4*sqrt(b*c)*a^3*c*weierstrassPInverse(-4*a/b, 0, x) - (7*b^3*c*x^4 + 13*a*b^2*c*x^2 + 4*a^2*b*c)*sqrt(b
*x^2 + a)*sqrt(c*x))/b^2

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Sympy [C] Result contains complex when optimal does not.
time = 2.80, size = 46, normalized size = 0.25 \begin {gather*} \frac {a^{\frac {3}{2}} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(3/2)*(b*x**2+a)**(3/2),x)

[Out]

a**(3/2)*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-3/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(9/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(3/2)*(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(3/2)*(c*x)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(3/2)*(a + b*x^2)^(3/2),x)

[Out]

int((c*x)^(3/2)*(a + b*x^2)^(3/2), x)

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